The Fused Lasso Under Linear Inequality

The predecessors already have done a lot about the question of the regression coefficients in the linear models. The estimation of the regression coefficients has many methods, including ordinary least squares, ridge regression and principal components regression. But they have a common shortcoming that they do not produce sparse models. In order to solve the problem, Tibshirani and Saunders(1996, 2005) proposed the lasso and the fused lasso which can work well in variable selection. This article considers that some prior conditions can be known in variable selection except for the sample information. We add the linear inequality restraint to improve the fused lasso. We not only propose the improved the fused lasso, but also propose a new Monte Carlo method to solve the problem. The Monte Carlo method behaves very well and costs less time. Afterwards, we propose the standard error and degrees of the freedom of the fused lasso under the linear inequality , and prove the asymptotic properties for it. At the end of the passage, we have simulated in the computer and obtained a good result, confirming that fused lasso under linear inequality can be good at variable selection.